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Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
The time taken for a population to double in size is called the doubling time. For the Malthusian model, this time, td, is determined from the algebraic equation Ptd=2P(0) which leads to P(0)ertd=2P(0)and henceertd=2
First Order Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
The time taken for a population to double in size is called the doubling time. For the Malthusian model, this time, td, is determined from the algebraic equation P(td) = 2P(0) which leads to P(O)ertd=2P(0)and henceertd=2. $$ P(O)e^{{rt_{d} }} = 2P(0)\,{\text{and hence}}\,\,e^{{rt_{d} }} = 2. $$
Human Populations
Published in Gary S. Moore, Kathleen A. Bell, Living with the Earth, 2018
Gary S. Moore, Kathleen A. Bell
This relationship is derived from the formula for calculating compound interest at the bank and is an approximation. It is useful to remember that a growth rate in any population compounds itself since the individuals entering the population will ultimately reproduce and add more people to that population. As the growth rate increases, the doubling time decreases. Populations from 8,000 BCE to the 1650 doubled about every 1,500 years as evidenced by the relatively slow rate of population increase (Figure 2.9). The population next doubled in 200 years, then 80 years, then 45 years, and 36 years when the world reached 4 billion people in 1975; it is now over 7,449,698,000 people in 2017. Viewing population growth in this way is useful because most everybody can understand what life would be like if the number of people in our particular location were doubled within a generation. A graphical presentation of doubling time versus the rate of natural increase is shown in Figure 2.13. A rate of 4.0 percent annual increase will double the population for a country in 18 years. The World Bank estimated Kenya to have such a rate in 1979, dropping to 3.7 percent in 1988 with a population of 23.3 million.15 The population in Niger in 2018 is 22, 311, 375 with an annual rate of increase of 3.88 percent and a TFR of 7.35, and its doubling rate has been increased to 20.5 years.16 Although this may be viewed as improvement, such an increase in so short a time will likely surpass the carrying capacity of that country and it will be incredibly challenged to build roads, houses, schools, and sanitation facilities to accommodate this many people in so short a time.
Optimization of microalgae growth for biofuel production using a new empirical dynamic model
Published in Biofuels, 2021
Ibifubara Humphrey, Michael A. C. Chendo, Abdulahi N. Njah, Dike I. Nwankwo
At a light intensity of 1000 lux, the maximum optical density achieved was 0.645 on the 20th day of cultivation under magenta light. The specific growth rate was 0.332 ± 0.001 per day and the doubling time was 2.08 ± 0.008 days. Cultivation under red light gave a maximum optical density of 0.464 on the 13th day with a specific growth rate of 0.0269 ± 0.201 per day and a doubling time of 2.58 ± 0.20 days. Under blue light the maximum optical density was 0.514 on the 17th day with a specific growth rate and doubling time of 0.285 ± 0.022 per day and 2.43 ± 0.019 days, respectively. At a light intensity of 3000 lux, the maximum optical density of 0.821 was achieved on the 17th day of cultivation under magenta light with a specific growth rate of 0.393 ± 0.003 per day and a doubling time of 1.76 ± 0.014 days. The maximum optical density achieved under red light was 0.643 on the 18th day with a specific growth rate and doubling time of 0.321 ± 0.006 per day and 2.15 ± 0.04 days, respectively. For cultivation under blue light the maximum optical density was 0.645 on the 17th day, while the specific growth rate was 0.332 ± 0.001 per day and the doubling time was 2.08 ± 0.008 days. At a light intensity of 5000 lux, the maximum optical density reached was 0.997 on day 14th under magenta light, which was the highest optical density obtained in the experiment. The specific growth rate was 0.512 ± 0.006 per day and the doubling time was 1.35 ± 0.018 day. Under red light the maximum optical density was 0.777 on day 16th, and the specific growth rate and doubling time were 0.375 ± 0.002 per day and 1.85 ± 0.014 days, respectively. The maximum optical density under blue light was 0.793 on the 18th day with a specific growth rate of 0.379 ± 0.005 per day and a doubling time of 1.82 ± 0.026 days. The maximum optical density achieved under sunlight was 0.799 on the 18th day of cultivation; the specific growth rate was 0.381 ± 0.007 per day and the doubling time was 1.81 ± 0.034 days. The maximum optical density reached under white light was 0.815 on the 18th day, while the specific growth rate and the doubling time were 0.092 ± 0.005 per day and 1.78 ± 0.021 days, respectively.